# How are the two definitions of B, as magnetic field and magnetic flux density, same?

My question, is primarily, based on this question, in which the accepted answer asserts that $$\vec B \,$$ has two definitions,

1. As magnetic field$$^*$$, using Lorentz force $$\vec F=q \vec v \times \vec B$$
2. As magnetic flux density, from $$\Phi=\int\vec B.d\vec S$$

My question is how do these two definitions, result in the same vector field?

$$*$$ I'm not able to define $$\vec B$$ as I could define $$\vec E$$, as force per unit charge. Is there one such definition for $$\vec B$$, or was it introduced, so that we could formulate our observations easily with such a vector field?

In laws of induction, we use B as flux density and while finding the Lorentz force, we use it as a vector field whose magnitude is given by Biot-Savart law. How are these two equal?

• I don't think the answer you refer to really does claim that these are both definitions. The relation involving $\Phi$ can't define $B$ unless you already have a definition of $\Phi$. – Ben Crowell Sep 14 at 13:51
• @BenCrowell Then I really don't understand that answer and my question is the same as that. In laws of induction, we use B as flux density and while finding the Lorentz force, we use it as a vector field whose magnitude is given by Biot-Savart law. How are these two equal? – Aravindh Vasu Sep 14 at 13:54